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\parbox{1.95in}{\bf Math 3110}
\hfill {\Large \bf Homework \#9} \hfill
\parbox{1.95in}{\bf \hfill Due: Mon., Nov. 18${}^\text{\bf th}$, 2024}

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Please remember when submitting any work via email or in person to\dots\\
\force \hfill \fbox{\fbox{\bf\large\color{red}PUT YOUR NAME ON YOUR WORK!}} \hfill \force


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\begin{description}
\item[\#1] In each of the following rings, $R$, state the characteristic of the ring. If $R$ has unity, 
     give an example of a unit and its inverse (other than $1$ itself). If no unit exists, explain why not. 
     If $R$ has zero divisors, give an example of a zero divisor. If no zero divisors exist, explain why not.
     \begin{enumerate}[(a)]
     \item $R = \mathbb{Z}_{6} \times \mathbb{Z}_{15} $
     \item $R = 5\mathbb{Z} = \{ 5k \,|\, k \in \mathbb{Z} \} $ (multiples of 5)
     \item $R = (\mathbb{Z}_{11})^{2 \times 2} $ ($2 \times 2$ matrices with entries in $ \mathbb{Z}_{11}$)
     \end{enumerate}     

\item[\#2] Let $R$ be a ring with $1$. Let $u$ be a unit and $z$ be a zero divisor. Prove that $zu$ is a zero divisor but $u^2$ is a unit.

{\it Note:} Argue carefully. You will need to write separate arguments for the cases where $z$ is a left zero divisor and where $z$ is a right zero divisor.

\item[\#3] Let $\mathbb{Z}[\sqrt{-5}] = \{ a+b\sqrt{-5} \;|\; a,b \in \mathbb{Z} \}$. 

This ring is equipped with an extra tool called an algebraic {\it norm}. We define the norm of $z=a+b\sqrt{-5}$ by $N(z)= (a+b\sqrt{-5})(a-b\sqrt{-5})=a^2+5b^2$. Notice that $N(z) \in \mathbb{Z}_{\geq 0}$.  

A very helpful observation is that for any $w,z \in \mathbb{Z}[\sqrt{-5}]$, we have $N(wz)=N(w)N(z)$.

   \begin{enumerate}[(a)]
   \item Prove $\mathbb{Z}[\sqrt{-5}]$ is a subring of $\mathbb{C}$.
   \item When is $N(z)=0$? When is $N(z)=1$?
   \item If $u \in U(\mathbb{Z}[\sqrt{-5}])$, then what can be said about $N(u)$? [{\it Hint:} Consider $N(uu^{-1})$.] Determine $U(\mathbb{Z}[\sqrt{-5}])$.
   \item Prove that $\mathbb{Z}[\sqrt{-5}]$ has no zero divisors. [{\it Hint:} Use the norm.] 
   \end{enumerate}
   
   {\it Note:} Since $\mathbb{Z}[\sqrt{-5}]$ is a commutative ring with $1\not=0$ and it has no zero divisors, it is an {\it integral domain}.
   
\item[\#4] Prove that $\mathbb{Q}[\sqrt{-5}] = \{ a+b\sqrt{-5} \;|\; a,b \in \mathbb{Q} \}$ is a field. 

{\it Note:} Use a subring test to show it is a ring (it is a subring of $\mathbb{C}$).\\ \force \hfill Then use the ``conjugate trick'' to show it has multiplicative inverses.

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